Vibrating tube densitometers are a well-known form of apparatus for measuring the density of a flowing medium. One example of this form of apparatus is described in British Patent 2 062 865.
In operation, a vibrating tube densitometer is excited so as to vibrate, in a particular mode, at its resonant frequency. This resonant frequency will be effected by changes in the density of the fluid contained in, or passing through, the tube. The indicated density will also be effected by the fluid temperature and/or fluid pressure to which the vibrating tube is subjected.
This requires each densitometer to be calibrated as can be more readily understood with reference to the following:
The resonant frequency of a vibrating tube densitometer with fluid contained in it can be expressed as:
                    f        =                              1                          2              ⁢              π                                ⁢                                    k                                                m                  r                                +                                                      V                    f                                    ·                                      ρ                    f                                                                                                          (                  Equation          ⁢                                          ⁢          1                )            where                f is the resonant frequency of the vibrating tube densitometer containing a fluid        mr is the mass of the resonant element within the vibrating tube densitometer        Vf is the volume of the fluid contained in the resonant element        ρf is the density of the fluid contained in the resonant element        k is the stiffness of the resonant element        
Among the above parameters, mr is a constant. All the other parameters vary with measurement conditions, i.e. mainly temperature (t) and pressure (p), therefore we have Vf(t, p), ρf(t, p), k(t, p)−fluid volume, fluid density and resonant element stiffness as functions of temperature and pressure respectively.
At measurement conditions, the resonant frequency (f) of a vibrating tube densitometer containing a fluid, varies with not only the fluid density ρf(t,p), but also with the volume of the fluid Vf(t, p) and the stiffness of the resonant element k(t, p) which both are affected by the temperature/pressure effects of the vibrating tube densitometer.
Equation 1 can be rewritten in terms of fluid density as:ρf=K0+K2τ2  (Equation 2)where K0=−mr/Vf, K2=k/(4π2Vf), and r=1/f is the period of oscillation.
As Equation 1 is only a first order approximation to the actual behavior of a vibrating tube densitometer containing a fluid, more generic equations have been developed for use in the calibration of specific vibration tube densitometers.
One example of such a generic equation is:D=K0+K1·τ+K2·τ2  (Equation 3)in which K0, K1, and K2 are density coefficients to be calibrated, D is the indicated fluid density, and τ is the period of oscillation.
One way to calibrate such a densitometer is to determine K0, K1, and K2, across the full operational temperature and pressure range, with fluids of known density at those conditions. The relationships between K0, K1, K2, and temperature and pressure, can then be derived. This method requires numerous calibration points.
One more conventional way to calibrate such a densitometer is to first determine density coefficients K0, K1, and K2 at a reference temperature and pressure condition, such as at temperature t0=20° C. and at atmospheric pressure p0=1 BarA; then determine the temperature effects of the densitometer at the reference pressure condition; and then determine the pressure effects of the densitometer at the reference temperature condition. In other words, the temperature effects of densitometer are calibrated at the reference pressure condition and the pressure effects of densitometer are calibrated at the reference temperature condition.
When a densitometer, so calibrated, operates at other temperatures and elevated pressures, the indicated fluid density is calculated first, and then corrected for the above temperature effects characterized at the reference pressure condition and for the above pressure effects characterized at the reference temperature condition. For example:
One form of temperature correction is:Dt=D·(1+K18·(t−t0))+K19·(t−t0)  (Equation 4)where t is the operating temperature, t0 is the reference temperature and K18 and K19 are temperature correction coefficient constants. The temperature correction coefficient constants K18 and K19 are generally calibrated at atmospheric pressure p0=1 BarA. If necessary or desired in a complex situation, K18 and K19 can be expressed as functions of temperature.
One form of pressure correction is:Dp=Dt·(1+K20·(p−p0))+K21·(p−p0)  (Equation 5)K20=K20A+K20B·(p−p0)+K20C·(p−p0)2  (Equation 6)K21=K21A+K21B·(p−p0)+K231C·(p−p0)2  (Equation 7)where p is the operating pressure, p0 is the reference pressure, and K20A, K20B, K20C, K21A, K21B and K21C are pressure correction coefficient constants. The pressure correction coefficient constants K20A, K20B, K20C, K21A, K21B and K21C are generally calibrated at a reference temperature t0=20° C. K20 and K21 can, if necessary or desired, be expanded as higher order polynomial functions of pressure, or expressed as other functions of pressure.
A problem with the above-described calibration is that, at combined elevated pressure and temperature, measurement errors may be observed between the corrected density value Dp and the true density of the fluid under measurement, By way of example, at a combined condition of 80° C. and 100 BarG on a fluid of base density 826.8 kg/m3, the measurement error can be as great as 0.25% or 2 kg/m3. This may exceed the error-acceptance level of many applications, particularly fiscal metering applications.
It is an object of this invention to provide a method of calibrating a vibrating tube densitometer which will go at least some way in addressing the problem described, or which will at least provide a novel and useful addition to the art.